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Register a new userOn Coresets for Logistic Regression
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Added by
Chris Schwiegelshohn
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Coresets are one of the central methods to facilitate the analysis of large data sets. We continue a recent line of research applying the theory of coresets to logistic regression. First, we show a negative result, namely, that no strongly sublinear sized coresets exist for logistic regression. To deal with intractable worst-case instances we introduce a complexity measure $mu(X)$, which quantifies the hardness of compressing a data set for logistic regression. $mu(X)$ has an intuitive statistical interpretation that may be of independent interest. For data sets with bounded $mu(X)$-complexity, we show that a novel sensitivity sampling scheme produces the first provably sublinear $(1pmvarepsilon)$-coreset. We illustrate the performance of our method by comparing to uniform sampling as well as to state of the art methods in the area. The experiments are conducted on real world benchmark data for logistic regression.
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For random field theory based multiple comparison corrections In brain imaging, it is often necessary to compute the distribution of the supremum of a random field. Unfortunately, computing the distribution of the supremum of the random field is not easy and requires satisfying many distributional assumptions that may not be true in real data. Thus, there is a need to come up with a different framework that does not use the traditional statistical hypothesis testing paradigm that requires to compute p-values. With this as a motivation, we can use a different approach called the logistic regression that does not require computing the p-value and still be able to localize the regions of brain network differences. Unlike other discriminant and classification techniques that tried to classify preselected feature vectors, the method here does not require any preselected feature vectors and performs the classification at each edge level.
With the rapid development of social media sharing, people often need to manage the growing volume of multimedia data such as large scale video classification and annotation, especially to organize those videos containing human activities. Recently, manifold regularized semi-supervised learning (SSL), which explores the intrinsic data probability distribution and then improves the generalization ability with only a small number of labeled data, has emerged as a promising paradigm for semiautomatic video classification. In addition, human action videos often have multi-modal content and different representations. To tackle the above problems, in this paper we propose multiview Hessian regularized logistic regression (mHLR) for human action recognition. Compared with existing work, the advantages of mHLR lie in three folds: (1) mHLR combines multiple Hessian regularization, each of which obtained from a particular representation of instance, to leverage the exploring of local geometry; (2) mHLR naturally handle multi-view instances with multiple representations; (3) mHLR employs a smooth loss function and then can be effectively optimized. We carefully conduct extensive experiments on the unstructured social activity attribute (USAA) dataset and the experimental results demonstrate the effectiveness of the proposed multiview Hessian regularized logistic regression for human action recognition.
Wasserstein distance-based distributionally robust optimization (DRO) has received much attention lately due to its ability to provide a robustness interpretation of various learning models. Moreover, many of the DRO problems that arise in the learning context admits exact convex reformulations and hence can be tackled by off-the-shelf solvers. Nevertheless, the use of such solvers severely limits the applicability of DRO in large-scale learning problems, as they often rely on general purpose interior-point algorithms. On the other hand, there are very few works that attempt to develop fast iterative methods to solve these DRO problems, which typically possess complicated structures. In this paper, we take a first step towards resolving the above difficulty by developing a first-order algorithmic framework for tackling a class of Wasserstein distance-based distributionally robust logistic regression (DRLR) problem. Specifically, we propose a novel linearized proximal ADMM to solve the DRLR problem, whose objective is convex but consists of a smooth term plus two non-separable non-smooth terms. We prove that our method enjoys a sublinear convergence rate. Furthermore, we conduct three different experiments to show its superb performance on both synthetic and real-world datasets. In particular, our method can achieve the same accuracy up to 800+ times faster than the standard off-the-shelf solver.
In streaming Singular Value Decomposition (SVD), $d$-dimensional rows of a possibly infinite matrix arrive sequentially as points in $mathbb{R}^d$. An $epsilon$-coreset is a (much smaller) matrix whose sum of square distances of the rows to any hyperplane approximates that of the original matrix to a $1 pm epsilon$ factor. Our main result is that we can maintain a $epsilon$-coreset while storing only $O(d log^2 d / epsilon^2)$ rows. Known lower bounds of $Omega(d / epsilon^2)$ rows show that this is nearly optimal. Moreover, each row of our coreset is a weighted subset of the input rows. This is highly desirable since it: (1) preserves sparsity; (2) is easily interpretable; (3) avoids precision errors; (4) applies to problems with constraints on the input. Previous streaming results for SVD that return a subset of the input required storing $Omega(d log^3 n / epsilon^2)$ rows where $n$ is the number of rows seen so far. Our algorithm, with storage independent of $n$, is the first result that uses finite memory on infinite streams. We support our findings with experiments on the Wikipedia dataset benchmarked against state-of-the-art algorithms.
We consider the setting of online logistic regression and consider the regret with respect to the 2-ball of radius B. It is known (see [Hazan et al., 2014]) that any proper algorithm which has logarithmic regret in the number of samples (denoted n) necessarily suffers an exponential multiplicative constant in B. In this work, we design an efficient improper algorithm that avoids this exponential constant while preserving a logarithmic regret. Indeed, [Foster et al., 2018] showed that the lower bound does not apply to improper algorithms and proposed a strategy based on exponential weights with prohibitive computational complexity. Our new algorithm based on regularized empirical risk minimization with surrogate losses satisfies a regret scaling as O(B log(Bn)) with a per-round time-complexity of order O(d^2).
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